A numerical method is presented for mesh-free calculation of moving interface problems in two-phase flow. In this method, the moving particle semi-implicit (MPS) method is combined with the meshless advection using flow-directional local-grid (MAFL) method, for an arbitrary Lagrangian-Eulerian calculation. Moving interfaces are directly traced in Lagrangian coordinates, while fixed boundaries such as inlet and outlet flows are calculated in Eulerian coordinates. The phase interface in two-phase flow is clearly calculated by tracing the computing points on the bubble surface. A calculation model for surface tension force is presented using the curvature of radius. The volume decrease in a gas bubble due to surface tension force is successfully computed. A kernel function with a variable kernel size is introduced to allow local concentration of computing points. A two-dimensional heat diffusion problem is solved using the variable kernel size model, and the result agrees with the exact solution. The void generation process in superheated liquid is simulated using the present method. The calculated bubble growth rate exactly coincides with the analytical solution. Rising bubble shapes in viscous liquid are also simulated in two dimensions, and the estimated shapes show good agreement with those of experiments.